RAROC-Kelly-Plus: smarter decisions

The risk-adjusted return on capital – RAROC – criterion for decision making was proposed by Bankers Trust We heard a lot about it in the 1990s, perhaps driven by consultancies. But as late as 2011 McKinsey's paper on The Use of Economic Capital suggested from their survey of banks that:

... the typical bank uses RAROC in a backward-looking manner and at the aggregate, not the transaction, level

But, adopted or not, vanilla RAROC is flawed; using it without adjustment can lead to writing business with sub-optimal and even negative returns.

Using simple examples we demonstrate the flaws and point to a more robust alternative and how it might be operationalised. That's RAROC-Kelly-Plus.

This is the full version of the summary article – the summary version goes beyond RAROC and contains a little less maths!

RAROC: the traditional solution to "shall we do A or B?"

The RAROC decision criterion maximises the ratio of two present values:

[1] The numerator: profits

The RAROC numerator is the value of expected future profits, discounted back to day-1 at a hurdle rate.

There are several different approaches to calculate such expected profit – for example traditional discounting at a high hurdle rate or a so-called market-consistent approach.

[2] The denominator: economic capital

The denominator is the capital a bank or insurer needs to set aside as an appropriate buffer against possible losses from a transaction – or indeed a business line or its operations overall.

So what's the problem?

I have often regretted the casual way the (actuarial) profession compresses the varied complexity of its affairs into a colourless present value. Frank Redington, voted the greatest British actuary ever

Not for the first time, Redington was right. Let's look at two flaws.

RAROC exposed – the two flaws

Sometimes RAROC is described as "optimising profits". In fact the two aspects are:

  1. Optimise capital: the weighted average cost of capital (WACC) is minimised through selecting appropriate mix of assets, debt and equity – the funding. This is a subtle, iterative and demanding process in which financial institutions has ploughed resource and money.
  2. Maximise expected profits: Here there has been less progress. Leaving aside the practical doubts raised by McKinsey, we suggest RAROC needs theoretical adjustment. Simple examples demonstrate the flaws in maximising the numerator – indeed the flaws are easier to miss with real world complexity.

The simple example – a fair coin

If RAROC is a good methodology it should result in good decisions in a simple coin tossing example. If not we need to look for improvements at this simple level, before generalising to the more complex business environment. So let's consider a sequence of coin tosses:

  • This coin is "fair" i.e. the probability of heads / tails are both 1/2.
  • The payoffs are known. A heads is a win – you are paid £W – while tails means you pay £L.

We'll drop the £ sign. Also assume that W and L are the potential returns per unit of investment. In this simplest of examples:

  • Economic capital is L: the maximum loss is L (per toss) meaning that the economic capital at any reasonable level (1-in-2 or 1-in-200) is also L.
  • Expected profit is 0.5 * (W - L): this follows immediately from the definition of (mathematical) expectation.
  • RAROC is 0.5 * (W - L) / L: from the RAROC definition.

Flaw 1: RAROC-based investments can be loss making

The RAROC approach fails to allow for how randomness compounds over time. It suggests maximising the RAROC ratio and, subject to this maximisation, going ahead if RAROC is greater than zero. In the simple case above, the bet would be taken if W - L > 0, so long as there was no better bet on offer.

Here's the loss problem. In our example one win and one loss results in compounding of (1 + W) * (1 - L) = 1 + W - L - W * L i.e. a return of W - L - W * L. The problem arises where W - L > 0, but W - L - W * L < 0 or, equivalently, L > 1 / (1 + W).

Take W = 24% and L = 20%:

  • The expected profit is 2% i.e. 0.5 * (W - L).
  • But over the long run expected compounding will dominate and (1 + W) * (1 - L) = 1.24 * 0.8 < 1.

That's RAROC giving us a decision we expect to lose money over the long run.

Flaw 2: RAROC gives no guidance on the amount to invest

Consider a bet where W = 2 and L = 1 and the probabilities are still 0.5. Suppose this is the only investment. What proportion of our wealth (or the insurer's capital) should be invested in it? Investing everything is foolhardy – there's a 50% probability of losing all the capital. But it's a good bet: 10%? 25%? RAROC is silent.

RAROC enhanced – the Kelly Criterion

Background: what is the Kelly Criterion?

The Kelly Criterion (1956) is named after a US scientist, John Kelly, of Bell Labs. The criterion gives a formula to determine what proportion of wealth to risk in a sequence of positive expected value bets so as to maximize the rate of growth of wealth. The Kelly approach is positive rather than normative; it does not say that that you should maximize the rate of growth, but rather that if you want to do so (without any other constraints) the Kelly approach is the way to do so.

Progress since the 1950s

The original "vanilla" Kelly Criterion applied only in limited circumstances – the typical example was coin tossing with known probabilities and payoffs – but the last 50 years have seen generalisations to encompass (e.g.) continuous distributions of returns and unknown probabilities.

The Criterion does not assume any particular utility function, but if the investor has a log utility function Kelly will return results which maximise expected utility.

Current attitudes to the Criterion

The original "vanilla" Kelly Criterion was (and is) somewhat controversial, particular among economists. Nobel prize winning economist Paul Samuelson disapproved so strongly that in 1979 he wrote the (almost) monosyllabic research paper Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long.

Samuelson has not has it all his own way; William T. Ziemba provided a detailed response in which he agreed with many of Samuelson's points, but not his conclusion. Ziemba implies that Professor Berlekamp – a key intellectual influence on arguably the world's most successful hedge fund was a Kelly fan.

Investors who behave as if they were full or close to full Kelly bettors include George Soros, Warren Buffett and John Maynard Keynes.William T Ziemba: Response to Paul Samuelson Letters and Papers on the Kelly Capital Growth Investment Strategy

The Criterion has gained more traction among (some) investors – famously Ed Thorp and arguably Warren Buffett – and (many) gamblers. Indeed many of the references are written by a niche fund managers or sports betting types. The Criterion has been the subject of many books and papers by Ed Thorp and others.

How the Kelly Criterion deals with the RAROC flaws

It does so directly:

  • Loss making investments: Kelly looks at compound returns taking into account full variability, not single period returns based on expected value.
  • RAROC gives no guidance on the amount to invest: the proportion of captial to invest is the result or output of the Kelly Criterion.

Finally although Kelly does not concern itself directly with economic capital – the Criterion works with RAROC's numerator – we simply need to make a division.

So what's the formula?

Consider the following coin tossing scenario, which – the dependency q = 1 - p – we can abbreviate to (p,W,L):

ResultProbabilityPayoffWhat happened?
Heads = win p W We win W
Tails = lose q = 1 - p L We lose L

We know from the RAROC flaws above that investing simply if the expected "one off" gain is positive i.e. p * W - q * L > 0 is a flawed strategy. Instead Kelly says we should invest a proportion f which maximises (1 + f * W)p * (1 - f * L)q. Equivalently this maximises p * log(1 + f * W) + q * log(1 - f * L). This is a relatively simple calculus problem, the solution to which is:

The Kelly Criterion

The Kelly Criterion says that to maximize the rate of growth we should invest a proportion f = [p * W - q * L] / [W * L].

  • The numerator is our expected gain – what gamblers call the "edge"
  • The denominator is a scaling factor – this gets us beyond the expected value approach

Implications and special cases

Simple odds. If we take L = 1 the bet can be described as "W-to-1 odds" (although conventionally that also requires p = 0.5). The Kelly recommendation is to bet a proportion [p * W - q ] / W – gamblers often abbreviate this to "edge over odds".

p = 0.5. Again, following the earlier theory, we require (1 + f * W)1/2 * (1 - f * L)1/2 > 1, which simplifies to f * [W - L - f * W * L] > 0 i.e. W - L - f * W * L > 0.

Factoring out the odds. For a (p,W*k,L*k) bet Kelly returns f = [p * W * k - q * L * k] / [ W * k * L * k] which simplifies to [p * W - q * L] / [ W * L * k]. We can see that the proportion to invest has been scaled down by a factor of k relative to the (p,W,L) bet. The dynamics of this bet are driven by the ratio W / L. We can take k = 1 / L, when the proportion corresponding to (p,W/L,1) will be [p * (W / L) - q] / [W / L]. It really comes back to edge over odds.

Simplifying the odds. Due to the above logic we could, without losing generality, simply consider a (p,W,1) bet, where the Kelly proportion is (p * W - q) / W or, equivalently, p - q / W. Perhaps not surprisingly, we always bet a proportion a little under p and the higher W the closer we get to p.

Making it operational: RAROC-Kelly-Plus

Adjustments are required to turn RAROC-Kelly into a fully working decision making tool. Five areas are described below.

Good news

Significant progress has been made in all the areas below, so that RAROC-Kelly-Plus is a workable concept. I am not aware, however, that it has yet been deployed in practice. Deployment would inevitably and advisably us a step-by-step approach:

  • Examine current decision making methodology: Does it use RAROC and modelling?
  • Flaws in RAROC-based decisions:
    1. Negative returns: is there anything in place that stops a project with negative expected compound returns going ahead? Has this happened in practice?
    2. Sub-optimal sizing: should we have taken a different level of nominal exposure? Typically this would involve taking a share rather than the whole deal.
  • Learning the lessons: happily we can learn lessons from this approach without a full RAROC-Kelly-Plus implementation – see below.

[1] Continuous variation

In most commercial situations gains and losses effectively follow a continuous distribution, where larger losses have smaller probabilities. The coin tossing is usually only illustrative. Suggesting that many investors would benefit from studying the Kelly Criterion, UK actuary and private investor Guy Thomas says:

It is not so much that it gives directly applicable answers, but more than it directs attention to the right questions.Source: Guy Thomas writing in Free Capital

It seems that this view is backed up strongly by Boyles asset management in this extract from their letter to shareholders.

[2] Uncapped losses

Insurers may enter into contracts which, in principle, have no finite upper limit. Necessarily (since the area under the probability curve is 1) the probabilities associated with the largest losses become very small (and are asymptotically zero). But although the economic capital associated with these is qualifiable, there is inn theory a probability – infinitesimal no doubt – that such a contract exhausts the entire capital of the company.

[3] Unknown probabilities

In most commercial applications probabilities need to be estimated, rather than being given or deduced from (e.g.) symmetry arguments. Typically the estimates comes through a combination of data analysis and expert judgement, perhaps backed by a formal Bayesian framework. Again this challenge has been tackled.

[4] Limited time horizons

Another way of putting this is that we need to incorporate risk appetite. Professor Samuelson was not the first to spot the point that most investors cannot afford to wait for the timescales or number of investment realisations that would result in the Kelly Criterion's asymptotic properties dominating other strategies.

[5] Simultaneous investments

Extensions of the Kelly Criterion exist to deal with multiple investments.

In practice we do not deploy a proportion of available capital as economic capital against a bet (opportunity, project etc), wait for the outcome then deploy another proportion against another opportunity. Instead opportunities present themselves either simultaneously or at overlapping time intervals.

We also need to think about:

  • Opportunity cost: Does taking an opportunity now mean that we may need to pass on a more favourable opportunity later?
  • Dependency: There are various types of dependency. Two investments might be:
    • directly dependent e.g. the two investments might both fail because the second required the success of the first.
    • indirectly dependent e.g. pension liabilities in two different schemes are both likely to be affected by general population longevity improvements.
    We might along the lines of financial economics, think an insurer's portfolio as a bundle of risky investments.
© 2014-2017: 4A Risk Management; a trading name of Transformaction Development Limited